Basis functions

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captioner <- function(caption) {
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library(tidyr)
library(dplyr)
library(plotly)
library(MetricGraph)

gets.graph.basis <- function(h, cont = TRUE){
  #F
  edge1 <- rbind(c(0,0), c(0,2))
  edge2 <- rbind(c(0,2), c(0,4))
  edge3 <- rbind(c(0,2), c(1,2))
  edge4 <- rbind(c(0,4), c(2,4))
  #U
  edge5 <- rbind(c(2,4), c(2,1))
  thetau <- seq(pi, 2*pi, length.out = 100)
  edge6 <- cbind(3+1*cos(thetau), 1+1*sin(thetau))
  edge7 <- rbind(c(4,1), c(4,4))
  #N
  edge8 <- rbind(c(4,1), c(4,0))
  edge9 <- rbind(c(4,4), c(6,0))
  edge10 <- rbind(c(6,0), c(6,4))
  #C
  thetac1 <- seq(pi, pi/2, length.out = 100)
  edge11 <- cbind(8+2*cos(thetac1), 2+2*sin(thetac1))
  thetac2 <- seq(pi, 3*pi/2, length.out = 100)
  edge12 <- cbind(8+2*cos(thetac2), 2+2*sin(thetac2))
  #T
  edge13 <- rbind(c(8,4), c(10,4))
  edge14 <- rbind(c(9,4), c(9,0))
  #I
  edge15 <- rbind(c(10,4), c(12,4))
  edge16 <- rbind(c(10,0), c(12,0))
  edge17 <- rbind(c(11,0), c(11,4))
  #O
  thetao1 <- seq(pi, 2*pi, length.out = 100)
  edge18 <- cbind(13+1*cos(thetao1), 1+1*sin(thetao1))
  thetao2 <- seq(0, pi, length.out = 100)
  edge19 <- cbind(13+1*cos(thetao2), 3+1*sin(thetao2))
  edge20 <- rbind(c(12,3), c(12,1))
  edge21 <- rbind(c(14,1), c(14,4))
  #N
  edge22 <- rbind(c(14,4), c(16,0))
  edge23 <- rbind(c(16,0), c(16,4))
  edge56 <- rbind(c(14,0), c(14,1))
  #S
  edge24 <- rbind(c(16,0), c(17,0))
  tethas1 <- seq(-pi/2, pi/2, length.out = 100)
  edge25 <- cbind(17+1*cos(tethas1), 1+1*sin(tethas1))
  thetas2 <- seq(3*pi/2, pi/2, length.out = 100)
  edge26 <- cbind(17+1*cos(thetas2), 3+1*sin(thetas2))
  edge27 <- rbind(c(17,4), c(18,4))
  #H
  edge28 <- rbind(c(0,4), c(0,6))
  edge29 <- rbind(c(0,6), c(0,8))
  edge30 <- rbind(c(0,6), c(2,6))
  edge31 <- rbind(c(2,4), c(2,8))
  #A
  edge32 <- rbind(c(2,4), c(3,8))
  edge33 <- rbind(c(3,8), c(4,4))
  edge34 <- rbind(c(2.5,6), c(3.5,6))
  #T
  edge35 <- rbind(c(3,8), c(6,8))
  edge36 <- rbind(c(5,8), c(5,4))
  #B
  edge37 <- rbind(c(8,8), c(8,4))
  thetab1 <- seq(-pi/2, pi/2, length.out = 100)
  edge38 <- cbind(9+1*cos(thetab1), 5+1*sin(thetab1))
  edge39 <- rbind(c(8,6), c(9,6))
  edge40 <- rbind(c(9,8), c(8,8))
  thetab2 <- seq(-pi/2, pi/2, length.out = 100)
  edge41 <- cbind(9+1*cos(thetab2), 7+1*sin(thetab2))
  #A
  edge42 <- rbind(c(10,4), c(11,8))
  edge43 <- rbind(c(11,8), c(12,4))
  edge44 <- rbind(c(10.5,6), c(11.5,6))
  #S
  
  #I
  edge45 <- rbind(c(14,4), c(16,4))
  edge46 <- rbind(c(15,4), c(15,8))
  edge47 <- rbind(c(14,8), c(16,8))
  #S
  edge48 <- rbind(c(16,4), c(17,4))
  edge49 <- cbind(17+1*cos(tethas1), 5+1*sin(tethas1))
  edge50 <- cbind(17+1*cos(thetas2), 7+1*sin(thetas2))
  edge51 <- rbind(c(17,8), c(18,8))
  #S
  edge52 <- rbind(c(13,8), c(14,8))
  edge53 <-cbind(13+1*cos(thetas2), 7+1*sin(thetas2))
  edge54 <- cbind(13+1*cos(tethas1), 5+1*sin(tethas1))
  edge55 <- rbind(c(12,4), c(13,4))
  
  edges <- list(edge1, edge2, edge3, edge4, edge5, edge6, edge7,
                edge8, edge9, edge10, edge11, edge12, edge13, edge14,
                edge15, edge16, edge17, edge18, edge19, edge20, edge21,
                edge22, edge23, edge24, edge25, edge26, edge27,
                edge28, edge29, edge30, edge31, edge32, edge33, edge34,
                edge35, edge36, edge37, edge38, edge39, edge40, edge41,
                edge42, edge43, edge44, edge45, edge46, edge47,
                edge48, edge49, edge50, edge51, edge52, edge53, edge54, edge55, edge56)
  graph <- metric_graph$new(edges = edges, perform_merges = TRUE)
  graph$prune_vertices()
  graph$build_mesh(h = h, continuous = cont)
  return(graph)
}

add_group_boundaries <- function(mat) {
  # Unique group identifiers in the first column
  groups <- unique(mat[, 1])
  
  # Initialize list to store results
  result_list <- vector("list", length(groups))
  
  for (i in seq_along(groups)) {
    grp <- groups[i]
    group_rows <- mat[mat[, 1] == grp, , drop = FALSE]
    
    # Add boundary rows
    augmented <- rbind(
      c(grp, 0),
      group_rows,
      c(grp, 1)
    )
    result_list[[i]] <- augmented
  }
  
  # Combine all groups
  result <- do.call(rbind, result_list)
  rownames(result) <- NULL
  return(result)
}

# Function to insert NA row between groups
insert_na_between_groups <- function(mat, group_vec) {
  # Split the matrix by group
  mat_split <- split(as.data.frame(mat), group_vec)
  
  # Add NA rows after each group
  with_na <- lapply(mat_split, function(x) rbind(as.matrix(x), rep(NA, ncol(mat))))
  
  # Combine everything into one matrix (removing the last NA if not needed)
  mat <- do.call(rbind, with_na)
  return(mat) #mat[-nrow(mat), ]
}

fill_between_NA <- function(vec) {
  
  # Find the indices of the NA values
  na_indices <- which(is.na(vec))
  
  for (i in seq_along(na_indices)[-length(na_indices)]) {
    start <- na_indices[i]
    end <- na_indices[i + 1]
    
    # Work on the elements between two NA values
    if (end - start > 1) {
      segment <- vec[(start + 1):(end - 1)]
      if (all(segment == 0, na.rm = TRUE)) {
        vec[(start + 1):(end - 1)] <- NA
      }
    }
  }
  return(vec)
}

keep_nonzeros_and_border_zeros <- function(vec) {
  n <- length(vec)
  keep <- rep(FALSE, n)
  
  # Identify nonzero values (ignoring NAs)
  is_nonzero <- !is.na(vec) & vec != 0
  
  for (i in which(is_nonzero)) {
    keep[i] <- TRUE
    if (i > 1 && !is.na(vec[i - 1]) && vec[i - 1] == 0) keep[i - 1] <- TRUE
    if (i < n && !is.na(vec[i + 1]) && vec[i + 1] == 0) keep[i + 1] <- TRUE
  }
  
  # Replace zeros that are not marked for keeping with NA
  vec[!is.na(vec) & vec == 0 & !keep] <- NA
  return(vec)
}

graph <- gets.graph.basis(h = 1/1, cont = TRUE)
graph_cont <- gets.graph.basis(h = 1/50, cont = TRUE)

# discontinuous mesh
V <- graph_cont$mesh$V
VtE <- graph_cont$mesh$VtE


new_VtE <- add_group_boundaries(VtE[(graph$nV+1):nrow(VtE),])
new_V <- graph_cont$coordinates(PtE = new_VtE, normalized = TRUE)
V_with_NA <- rbind(c(NA,NA), insert_na_between_groups(new_V, new_VtE[,1]))
x <- V_with_NA[,1]
y <- V_with_NA[,2]


A <- as.matrix(graph$fem_basis(new_VtE))
A_with_NA <- rbind(rep(NA, ncol(A)), insert_na_between_groups(A, new_VtE[,1]))
A_with_NA_cleaned <- apply(A_with_NA, 2, fill_between_NA)
A_with_NA_zeroed <- apply(A_with_NA, 2, keep_nonzeros_and_border_zeros)

x_range <- range(x, na.rm = TRUE)
y_range <- range(y, na.rm = TRUE)
z_range <- c(0,1)

# Get all z values for vertical lines
z_vals <- apply(A_with_NA_zeroed, 1, max, na.rm = TRUE)
# Subsample every 5th index
idx <- seq(1, nrow(A_with_NA_zeroed), by = 20)

# Subsample x, y, and z for gray lines
Z_red <- unlist(lapply(z_vals[idx], function(zj) c(0, zj, NA)))
X_red <- rep(x[idx], each = 3)
Y_red <- rep(y[idx], each = 3)

# Start plot
p <- plot_ly() %>% 
  add_trace(x = rep(x, times = graph$nV), 
            y = rep(y, times = graph$nV), 
            z = as.vector(A_with_NA_zeroed[, 1:graph$nV]), 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "red", width = 2)) %>%
  add_trace(x = X_red, y = Y_red, z = Z_red,
            type = "scatter3d", mode = "lines",
            line = list(color = "gray", width = 0.5),
            showlegend = FALSE) %>%
  add_trace(x = rep(x, times = ncol(A_with_NA_zeroed) - graph$nV), 
            y = rep(y, times = ncol(A_with_NA_zeroed) - graph$nV), 
            z = as.vector(A_with_NA_zeroed[, (graph$nV+1):ncol(A_with_NA_zeroed)]), 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "blue", width = 2)) %>%
  add_trace(x = x, 
            y = y, 
            z = x*0, 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "black", width = 4)) %>% 
  add_trace(x = rep(x, times = graph$nV), 
            y = rep(y, times = graph$nV), 
            z = as.vector(A_with_NA_zeroed[, 1:graph$nV])*0, 
            type = "scatter3d",
            mode = "lines", 
            showlegend = FALSE, 
            line = list(color = "green", width = 4)) %>%
  layout(scene = list(
    xaxis = list(title = "x", range = x_range),
    yaxis = list(title = "y", range = y_range),
    zaxis = list(title = "z", range = z_range),
    aspectratio = list(x = 2.4, y = 1.2, z = 0.06),
    camera = list(eye = list(x = -2, y = -3, z = 1.5), 
                  center = list(x = 0, y = 0, z = 0))))

1 Illustration

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p

Figure 1: Illustration of the basis function system \(\{\psi^i_h\}_{i=1}^{N_h}\) on the hat-basis-functions graph (in black). Standard hat functions associated with internal edge nodes are shown in blue, while special vertex-centered functions are highlighted in red.

1.1 References

grateful::cite_packages(output = "paragraph", out.dir = ".")

We used R version 4.5.0 (R Core Team 2025) and the following R packages: gsignal v. 0.3.7 (Van Boxtel, G.J.M., et al. 2021), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), knitr v. 1.50 (Xie 2014, 2015, 2025), Matrix v. 1.7.3 (Bates, Maechler, and Jagan 2025), MetricGraph v. 1.5.0.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), patchwork v. 1.3.1 (Pedersen 2025), plotly v. 4.10.4 (Sievert 2020), RColorBrewer v. 1.1.3 (Neuwirth 2022), renv v. 1.0.7 (Ushey and Wickham 2024), reshape2 v. 1.4.4 (Wickham 2007), rmarkdown v. 2.29 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2024), rSPDE v. 2.5.1.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.4.0 (Wickham, Pedersen, and Seidel 2025), tidyverse v. 2.0.0 (Wickham et al. 2019), viridisLite v. 0.4.2 (Garnier et al. 2023), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

Aden-Buie, Garrick, and Matthew T. Warkentin. 2024. xaringanExtra: Extras and Extensions for “xaringan” Slides. https://doi.org/10.32614/CRAN.package.xaringanExtra.

Allaire, JJ, Yihui Xie, Christophe Dervieux, Jonathan McPherson, Javier Luraschi, Kevin Ushey, Aron Atkins, et al. 2024. rmarkdown: Dynamic Documents for r. https://github.com/rstudio/rmarkdown.

Bates, Douglas, Martin Maechler, and Mikael Jagan. 2025. Matrix: Sparse and Dense Matrix Classes and Methods. https://doi.org/10.32614/CRAN.package.Matrix.

Bolin, David, and Kristin Kirchner. 2020. “The Rational SPDE Approach for Gaussian Random Fields with General Smoothness.” Journal of Computational and Graphical Statistics 29 (2): 274–85. https://doi.org/10.1080/10618600.2019.1665537.

Bolin, David, Mihály Kovács, Vivek Kumar, and Alexandre B. Simas. 2024. “Regularity and Numerical Approximation of Fractional Elliptic Differential Equations on Compact Metric Graphs.” Mathematics of Computation 93 (349): 2439–72. https://doi.org/10.1090/mcom/3929.

Bolin, David, and Alexandre B. Simas. 2023. rSPDE: Rational Approximations of Fractional Stochastic Partial Differential Equations. https://CRAN.R-project.org/package=rSPDE.

Bolin, David, Alexandre B. Simas, and Jonas Wallin. 2023a. MetricGraph: Random Fields on Metric Graphs. https://CRAN.R-project.org/package=MetricGraph.

———. 2023b. “Statistical Inference for Gaussian Whittle-Matérn Fields on Metric Graphs.” arXiv Preprint arXiv:2304.10372. https://doi.org/10.48550/arXiv.2304.10372.

———. 2024. “Gaussian Whittle-Matérn Fields on Metric Graphs.” Bernoulli 30 (2): 1611–39. https://doi.org/10.3150/23-BEJ1647.

———. 2025. “Markov Properties of Gaussian Random Fields on Compact Metric Graphs.” Bernoulli. https://doi.org/10.48550/arXiv.2304.03190.

Bolin, David, Alexandre B. Simas, and Zhen Xiong. 2024. “Covariance-Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference.” Journal of Computational and Graphical Statistics 33 (1): 64–74. https://doi.org/10.1080/10618600.2023.2231051.

Cheng, Joe, Carson Sievert, Barret Schloerke, Winston Chang, Yihui Xie, and Jeff Allen. 2024. htmltools: Tools for HTML. https://doi.org/10.32614/CRAN.package.htmltools.

Garnier, Simon, Ross, Noam, Rudis, Robert, Camargo, et al. 2023. viridis(Lite) - Colorblind-Friendly Color Maps for r. https://doi.org/10.5281/zenodo.4678327.

Müller, Kirill. 2020. here: A Simpler Way to Find Your Files. https://doi.org/10.32614/CRAN.package.here.

Neuwirth, Erich. 2022. RColorBrewer: ColorBrewer Palettes. https://doi.org/10.32614/CRAN.package.RColorBrewer.

Pedersen, Thomas Lin. 2025. patchwork: The Composer of Plots. https://doi.org/10.32614/CRAN.package.patchwork.

R Core Team. 2025. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Sievert, Carson. 2020. Interactive Web-Based Data Visualization with r, Plotly, and Shiny. Chapman; Hall/CRC. https://plotly-r.com.

Ushey, Kevin, and Hadley Wickham. 2024. renv: Project Environments. https://doi.org/10.32614/CRAN.package.renv.

Van Boxtel, G.J.M., et al. 2021. gsignal: Signal Processing. https://github.com/gjmvanboxtel/gsignal.

Wickham, Hadley. 2007. “Reshaping Data with the reshape Package.” Journal of Statistical Software 21 (12): 1–20. http://www.jstatsoft.org/v21/i12/.

Wickham, Hadley, Mara Averick, Jennifer Bryan, Winston Chang, Lucy D’Agostino McGowan, Romain François, Garrett Grolemund, et al. 2019. “Welcome to the tidyverse.” Journal of Open Source Software 4 (43): 1686. https://doi.org/10.21105/joss.01686.

Wickham, Hadley, Thomas Lin Pedersen, and Dana Seidel. 2025. scales: Scale Functions for Visualization. https://doi.org/10.32614/CRAN.package.scales.

Xie, Yihui. 2014. “knitr: A Comprehensive Tool for Reproducible Research in R.” In Implementing Reproducible Computational Research, edited by Victoria Stodden, Friedrich Leisch, and Roger D. Peng. Chapman; Hall/CRC.

———. 2015. Dynamic Documents with R and Knitr. 2nd ed. Boca Raton, Florida: Chapman; Hall/CRC. https://yihui.org/knitr/.

———. 2025. knitr: A General-Purpose Package for Dynamic Report Generation in R. https://yihui.org/knitr/.

Xie, Yihui, J. J. Allaire, and Garrett Grolemund. 2018. R Markdown: The Definitive Guide. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown.

Xie, Yihui, Christophe Dervieux, and Emily Riederer. 2020. R Markdown Cookbook. Boca Raton, Florida: Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown-cookbook.